Random Waves and Localization

848 NOTICES OF THE AMS VOLUME 42, NUMBER 8 T his essay aims to give a brief and informal survey of recent work on Anderson localization. In this subject the problems are easy to state, and most of them are open. Throughout history mathematicians have had considerable experience of waves: the electromagnetic waves with which we see the world, the sound waves with which we hear language and music, and the electron waves that glue the world together. However, it was only in 1747 that d’Alembert wrote down the first wave equation, and it took a good part of a century for mathematicians to understand its solutions. This development brought an understanding of wave motion in a space that is empty or at least homogeneous in some sense. It is natural to ask about wave motion in a space that is not empty, where small random inhomogeneities can scatter the waves. Even today relatively little is known about this situation. The mathematical context of the description of wave motion is that of conservative linear wave equations. In most elementary accounts waves propagate in empty space, coasting along until they hit an obstacle or inhomogeneity located in some bounded region. At this point they scatter and take some other direction of propagation in the empty space. This is not always a realistic model of the true situation. For instance, the sound wave or the light wave may propagate in an atmosphere in which the local propagation speed varies from point to point. The corresponding classical wave equation is